In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$.

Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$. $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

A [i]unitary [/i] divisor d of a number $n$ is a divisor $n$ that has the property $\gcd (d, n/d) = 1$. If $n = 1620$, what is the sum of all of the unitary divisors of $d$?

Positive integers $k, m, n$ satisfy $mn=k^2+k+3$, prove that at least one of the equations $x^2+11y^2=4m$ and $x^2+11y^2=4n$ has an odd solution.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$ $ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$

For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

Source: 1976 Euclid Part A Problem 3 ----- The minimum value of the function $2x^2+6x+7$ is $\textbf{(A) } 7 \qquad \textbf{(B) } \frac{5}{2} \qquad \textbf{(C) } \frac{9}{4} \qquad \textbf{(D) } -\frac{9}{2} \qquad \textbf{(E) } \frac{5}{4}$

The $2013$ numbers $$\frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\frac{1}{2013 \times 2014}$$ are arranged randomly on a circle. (a) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$ . (b) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$ .

On a straight line, three distinct points $ M $, $ D $, $ H $ are given. Construct a right-angled triangle for which $ M $ is the midpoint of the hypotenuse, $ D $ is the point of intersection of the bisector of the right angle with the hypotenuse, and $ H $ is the foot of the altitude to the hypotenuse.

Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$.

The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$. [img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]

While Steve and LeRoy are fishing $ 1$ mile from shore, their boat springs a leak, and water comes in at a constant rate of $ 10$ gallons per minute. The boat will sink if it takes in more than $ 30$ gallons of water. Steve starts rowing toward the shore at a constant rate of $ 4$ miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

Find all positive integers $m$ for which there exist three positive integers $a,b,c$ such that $abcm=1+a^2+b^2+c^2$.

Let $a_1,a_2,\dots,a_{100}$ be a sequence of integers. Initially, $a_1=1$, $a_2=-1$ and the remaining numbers are $0$. After every second, we perform the following process on the sequence: for $i=1,2,\dots,99$, replace $a_i$ with $a_i+a_{i+1}$, and replace $a_{100}$ with $a_{100}+a_1$. (All of this is done simultaneously, so each new term is the sum of two terms of the sequence from before any replacements.) Show that for any integer $M$, there is some index $i$ and some time $t$ for which $|a_i|>M$ at time $t$.

Each vertex of a right $n$-gon $(n\geq 3)$ is colored in yellow, blue or red. On each turn are chosen two adjacent vertices in different color and then are recolored in the third. For which $n$ can we get from an arbitrary coloring of the $n$-gon a monochromatic one (in one color)?

Let the incircle of triangle $ ABC$ touch sides $ BC,\, CA$ and $ AB$ at $ D,\, E$ and $ F,$ respectively. Let $ \omega,\,\omega_{1},\,\omega_{2}$ and $ \omega_{3}$ denote the circumcircles of triangle $ ABC,\, AEF,\, BDF$ and $ CDE$ respectively. Let $ \omega$ and $ \omega_{1}$ intersect at $ A$ and $ P,\,\omega$ and $ \omega_{2}$ intersect at $ B$ and $ Q,\,\omega$ and $ \omega_{3}$ intersect at $ C$ and $ R.$ $ a.$ Prove that $ \omega_{1},\,\omega_{2}$ and $ \omega_{3}$ intersect in a common point. $ b.$ Show that $ PD,\, QE$ and $ RF$ are concurrent.

Prove that, we can't find positive numbers $m$ and $n$ such that, $$m^2+(m+1)^2=n^4+(n+1)^4$$

In an acute triangle $\triangle ABC$, the midpoint of $BC$ is $M$. Perpendicular lines $BE$ and $CF$ are drawn respectively on $AC$ from $B$ and on $AB$ from $C$ such that $E$ and $F$ lie on $AC$ and $AB$ respectively. The midpoint of $EF$ is $N.$ $MN$ intersects $AB$ at $K.$ Prove that, the four points $B,K,E,M$ lie on the same circle.

Find the minimum value of $ f(a)=\int_{0}^{1}x|x-a|\ dx$.

$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition: (1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$ (2) $d \mid (x_1+x_2+ \cdots x_n)$ Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.

Let $ABCD$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle AOB = \angle COD = 135^\circ$, $BC=1$. Let $B^\prime$ and $C^\prime$ be the reflections of $A$ across $BO$ and $CO$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $AB^\prime C^\prime$ and $BCD$, respectively. If $M$ is the midpoint of $OH_1$, and $O^\prime$ is the reflection of $O$ about the midpoint of $MH_2$, compute $OO^\prime$.

A rectangle is decomposed by $6$ vertical lines and $6$ horizontal lines in the $49$ small rectangles (see figure). The perimeter of each small rectangle is known to be a whole number of meters. In this case, will the perimeter of the large rectangle be a whole number of meters? [asy] unitsize(0.8 cm); draw((0,0)--(8.6,0)--(8.6,4.2)--(0,4.2)--cycle, linewidth(1.5*bp)); draw((0.7,0)--(0.7,4.2)); draw((1.2,0)--(1.2,4.2)); draw((4.1,0)--(4.1,4.2)); draw((5.6,0)--(5.6,4.2)); draw((6.0,0)--(6.0,4.2)); draw((7.4,0)--(7.4,4.2)); draw((0,0.4)--(8.6,0.4)); draw((0,1.0)--(8.6,1.0)); draw((0,1.5)--(8.6,1.5)); draw((0,2.5)--(8.6,2.5)); draw((0,3.1)--(8.6,3.1)); draw((0,3.5)--(8.6,3.5)); [/asy]

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

A line meets a segment $AB$ at point $C$. Which is the maximal number of points $X$ of this line such that one of angles $AXC$ and $BXC$ is equlal to a half of the second one?